According to the chain rule (see e.g. [1]}), we derive
\(\begin{split}&\Big |{\frac{\mathrm {d}^{r}}{\mathrm {d}t^{r}}}\log \widetilde{\varphi }_j(tzV_n^{-1})\Big |\\&\quad =\Big |r!\sum _{*(k_\cdot ,r,\cdot )} (-1)^{u(k_\cdot )-1}(u(k_\cdot )-1)!\widetilde{\varphi }_j(tzV_n^{-1})^{-u(k_\cdot )} \prod _{l=1}^{r} \frac{1}{k_l!} \Big (\frac{1}{l!}{\frac{\mathrm {d}^{l}}{\mathrm {d}t^{l}}}\widetilde{\varphi }(tzV_n^{-1})\Big )^{k_l} \Big |\\&\quad \le c(r)\sum _{*(k_\cdot ,r,\cdot )} \prod _{l=1}^{r}\big (z^lV_n^{-l}|X_j|^l\big )^{k_l}= c(r)z^rV_n^{-r}|X_j|^r\end{split}\) 
